LMFDB - Embedded newform 936.2.bi.b.361.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [936,2,Mod(361,936)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(936, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("936.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 936 = 2^{3} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	936.bi (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$7.47399762919$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.195105024.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 5x^{6} + 4x^{5} - 20x^{4} + 12x^{3} + 45x^{2} - 108x + 81 $ x^8 - 4x^7 + 5x^6 + 4x^5 - 20x^4 + 12x^3 + 45x^2 - 108*x + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			361.3
Root			$0.560908 + 1.63871i$ of defining polynomial
Character	$\chi$	$=$	936.361
Dual form			936.2.bi.b.433.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.61023i q^{5} +(-0.971521 - 0.560908i) q^{7} +(-0.971521 + 0.560908i) q^{11} +(-3.53796 - 0.694883i) q^{13} +(-2.77743 + 4.81064i) q^{17} +(-1.45204 - 0.838335i) q^{19} +(-1.63871 - 2.83834i) q^{23} -1.81333 q^{25} +(-0.167192 - 0.289586i) q^{29} +0.129717i q^{31} +(1.46410 - 2.53590i) q^{35} +(-3.92356 + 2.26527i) q^{37} +(-4.96410 + 2.86603i) q^{41} +(-2.24895 + 3.89529i) q^{43} -10.7985i q^{47} +(-2.87076 - 4.97231i) q^{49} -4.14771 q^{53} +(-1.46410 - 2.53590i) q^{55} +(0.549538 + 0.317276i) q^{59} +(-2.35387 + 4.07702i) q^{61} +(1.81381 - 9.23490i) q^{65} +(-12.4372 + 7.18062i) q^{67} +(7.45204 + 4.30244i) q^{71} +9.94462i q^{73} +1.25847 q^{77} +13.5970 q^{79} +5.75637i q^{83} +(-12.5569 - 7.24974i) q^{85} +(-12.4235 + 7.17272i) q^{89} +(3.04743 + 2.65956i) q^{91} +(2.18825 - 3.79016i) q^{95} +(7.80107 + 4.50395i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{7} - 6 q^{11} + 6 q^{13} - 6 q^{19} - 2 q^{23} - 20 q^{25} + 8 q^{29} - 16 q^{35} - 24 q^{37} - 12 q^{41} + 6 q^{43} + 2 q^{49} - 20 q^{53} + 16 q^{55} - 18 q^{59} - 4 q^{61} - 14 q^{65} - 42 q^{67}+ \cdots + 30 q^{97}+O(q^{100}) $ 8 * q - 6 * q^7 - 6 * q^11 + 6 * q^13 - 6 * q^19 - 2 * q^23 - 20 * q^25 + 8 * q^29 - 16 * q^35 - 24 * q^37 - 12 * q^41 + 6 * q^43 + 2 * q^49 - 20 * q^53 + 16 * q^55 - 18 * q^59 - 4 * q^61 - 14 * q^65 - 42 * q^67 + 54 * q^71 + 60 * q^77 + 16 * q^79 + 6 * q^85 + 18 * q^89 - 46 * q^91 - 16 * q^95 + 30 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/936\mathbb{Z}\right)^\times$.

$n$	$145$	$209$	$469$	$703$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$			2.61023i			1.16733i	0.811994	+	0.583666i	$0.198382\pi$
$5$			2.61023i			1.16733i	−0.811994	+	0.583666i	$0.801618\pi$
$6$		0			0
$7$	−0.971521	−	0.560908i	−0.367200	−	0.212003i	0.305034	−	0.952341i	$-0.401332\pi$
$7$	−0.971521	−	0.560908i	−0.367200	−	0.212003i	−0.672235	+	0.740338i	$0.734665\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.971521	+	0.560908i	−0.292925	+	0.169120i	−0.639260	−	0.768991i	$-0.720759\pi$
$11$	−0.971521	+	0.560908i	−0.292925	+	0.169120i	0.346335	+	0.938111i	$0.387426\pi$
$12$		0			0
$13$	−3.53796	−	0.694883i	−0.981253	−	0.192726i
$13$	−3.53796	−	0.694883i	−0.981253	−	0.192726i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.77743	+	4.81064i	−0.673625	+	1.16675i	0.303244	+	0.952913i	$0.401930\pi$
$17$	−2.77743	+	4.81064i	−0.673625	+	1.16675i	−0.976869	+	0.213840i	$0.931403\pi$
$18$		0			0
$19$	−1.45204	−	0.838335i	−0.333121	−	0.192327i	0.324105	−	0.946021i	$-0.394937\pi$
$19$	−1.45204	−	0.838335i	−0.333121	−	0.192327i	−0.657226	+	0.753694i	$0.728270\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.63871	−	2.83834i	−0.341695	−	0.591834i	0.643052	−	0.765822i	$-0.277668\pi$
$23$	−1.63871	−	2.83834i	−0.341695	−	0.591834i	−0.984748	+	0.173988i	$0.944334\pi$
$24$		0			0
$25$	−1.81333			−0.362665
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.167192	−	0.289586i	−0.0310468	−	0.0537747i	0.850084	−	0.526646i	$-0.176551\pi$
$29$	−0.167192	−	0.289586i	−0.0310468	−	0.0537747i	−0.881131	+	0.472872i	$0.843217\pi$
$30$		0			0
$31$			0.129717i			0.0232978i	0.999932	+	0.0116489i	$0.00370805\pi$
$31$			0.129717i			0.0232978i	−0.999932	+	0.0116489i	$0.996292\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.46410	−	2.53590i	0.247478	−	0.428645i
$36$		0			0
$37$	−3.92356	+	2.26527i	−0.645029	+	0.372408i	−0.786549	−	0.617528i	$-0.788134\pi$
$37$	−3.92356	+	2.26527i	−0.645029	+	0.372408i	0.141520	+	0.989935i	$0.454801\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.96410	+	2.86603i	−0.775262	+	0.447598i	−0.834749	−	0.550631i	$-0.814387\pi$
$41$	−4.96410	+	2.86603i	−0.775262	+	0.447598i	0.0594862	+	0.998229i	$0.481054\pi$
$42$		0			0
$43$	−2.24895	+	3.89529i	−0.342961	+	0.594027i	−0.984981	−	0.172661i	$-0.944763\pi$
$43$	−2.24895	+	3.89529i	−0.342961	+	0.594027i	0.642020	+	0.766688i	$0.278097\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	10.7985i		−	1.57512i	−0.616237	−	0.787561i	$-0.711344\pi$
$47$		−	10.7985i		−	1.57512i	0.616237	−	0.787561i	$-0.288656\pi$
$48$		0			0
$49$	−2.87076	−	4.97231i	−0.410109	−	0.710330i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.14771			−0.569732			−0.284866	−	0.958567i	$-0.591949\pi$
$53$	−4.14771			−0.569732			−0.284866	+	0.958567i	$0.591949\pi$
$54$		0			0
$55$	−1.46410	−	2.53590i	−0.197419	−	0.341940i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.549538	+	0.317276i	0.0715438	+	0.0413058i	0.535345	−	0.844633i	$-0.320182\pi$
$59$	0.549538	+	0.317276i	0.0715438	+	0.0413058i	−0.463801	+	0.885939i	$0.653515\pi$
$60$		0			0
$61$	−2.35387	+	4.07702i	−0.301382	+	0.522009i	−0.976449	−	0.215748i	$-0.930781\pi$
$61$	−2.35387	+	4.07702i	−0.301382	+	0.522009i	0.675067	+	0.737756i	$0.264115\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.81381	−	9.23490i	0.224975	−	1.14545i
$66$		0			0
$67$	−12.4372	+	7.18062i	−1.51945	+	0.877252i	−0.519708	+	0.854344i	$0.673959\pi$
$67$	−12.4372	+	7.18062i	−1.51945	+	0.877252i	−0.999738	+	0.0229086i	$0.992707\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	7.45204	+	4.30244i	0.884394	+	0.510605i	0.872105	−	0.489319i	$-0.162755\pi$
$71$	7.45204	+	4.30244i	0.884394	+	0.510605i	0.0122896	+	0.999924i	$0.496088\pi$
$72$		0			0
$73$			9.94462i			1.16393i	0.813214	+	0.581965i	$0.197716\pi$
$73$			9.94462i			1.16393i	−0.813214	+	0.581965i	$0.802284\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.25847			0.143416
$78$		0			0
$79$	13.5970			1.52978			0.764889	−	0.644162i	$-0.222794\pi$
$79$	13.5970			1.52978			0.764889	+	0.644162i	$0.222794\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			5.75637i			0.631843i	0.948785	+	0.315922i	$0.102314\pi$
$83$			5.75637i			0.631843i	−0.948785	+	0.315922i	$0.897686\pi$
$84$		0			0
$85$	−12.5569	−	7.24974i	−1.36199	−	0.786344i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−12.4235	+	7.17272i	−1.31689	+	0.760307i	−0.983227	−	0.182386i	$-0.941618\pi$
$89$	−12.4235	+	7.17272i	−1.31689	+	0.760307i	−0.333663	+	0.942692i	$0.608285\pi$
$90$		0			0
$91$	3.04743	+	2.65956i	0.319458	+	0.278798i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.18825	−	3.79016i	0.224510	−	0.388863i
$96$		0			0
$97$	7.80107	+	4.50395i	0.792079	+	0.457307i	0.840694	−	0.541511i	$-0.182147\pi$
$97$	7.80107	+	4.50395i	0.792079	+	0.457307i	−0.0486151	+	0.998818i	$0.515481\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.44462	+	2.50215i	0.143745	+	0.248974i	0.928904	−	0.370321i	$-0.120752\pi$
$101$	1.44462	+	2.50215i	0.143745	+	0.248974i	−0.785159	+	0.619294i	$0.787419\pi$
$102$		0			0
$103$	5.22047			0.514388			0.257194	−	0.966360i	$-0.417202\pi$
$103$	5.22047			0.514388			0.257194	+	0.966360i	$0.417202\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.91614	+	5.05090i	0.281914	+	0.488289i	0.971856	−	0.235575i	$-0.0756974\pi$
$107$	2.91614	+	5.05090i	0.281914	+	0.488289i	−0.689942	+	0.723865i	$0.742364\pi$
$108$		0			0
$109$		−	2.92820i		−	0.280471i	−0.990118	−	0.140236i	$-0.955214\pi$
$109$		−	2.92820i		−	0.280471i	0.990118	−	0.140236i	$-0.0447860\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	2.44304	−	4.23147i	0.229822	−	0.398064i	−0.727933	−	0.685648i	$-0.759519\pi$
$113$	2.44304	−	4.23147i	0.229822	−	0.398064i	0.957755	+	0.287585i	$0.0928522\pi$
$114$		0			0
$115$	7.40872	−	4.27743i	0.690867	−	0.398872i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	5.39666	−	3.11576i	0.494711	−	0.285621i
$120$		0			0
$121$	−4.87076	+	8.43641i	−0.442797	+	0.766946i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			8.31797i			0.743982i
$126$		0			0
$127$	−5.15977	−	8.93699i	−0.457856	−	0.793030i	0.540992	−	0.841028i	$-0.318049\pi$
$127$	−5.15977	−	8.93699i	−0.457856	−	0.793030i	−0.998847	+	0.0479985i	$0.984716\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	0.110761			0.00967723			0.00483861	−	0.999988i	$-0.498460\pi$
$131$	0.110761			0.00967723			0.00483861	+	0.999988i	$0.498460\pi$
$132$		0			0
$133$	0.940458	+	1.62892i	0.0815480	+	0.141245i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.02106	−	3.47626i	−0.514414	−	0.296997i	0.220232	−	0.975447i	$-0.429318\pi$
$137$	−6.02106	−	3.47626i	−0.514414	−	0.296997i	−0.734646	+	0.678451i	$0.762652\pi$
$138$		0			0
$139$	5.21515	−	9.03291i	0.442344	−	0.766161i	−0.555519	−	0.831504i	$-0.687481\pi$
$139$	5.21515	−	9.03291i	0.442344	−	0.766161i	0.997863	+	0.0653421i	$0.0208139\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3.82697	−	1.30938i	0.320027	−	0.109495i
$144$		0			0
$145$	0.755887	−	0.436411i	0.0627730	−	0.0362420i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	8.31587	+	4.80117i	0.681262	+	0.393327i	0.800330	−	0.599559i	$-0.204657\pi$
$149$	8.31587	+	4.80117i	0.681262	+	0.393327i	−0.119068	+	0.992886i	$0.537991\pi$
$150$		0			0
$151$		−	13.7267i		−	1.11706i	−0.829484	−	0.558531i	$-0.811365\pi$
$151$		−	13.7267i		−	1.11706i	0.829484	−	0.558531i	$-0.188635\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.338591			−0.0271963
$156$		0			0
$157$	8.96200			0.715245			0.357623	−	0.933866i	$-0.383587\pi$
$157$	8.96200			0.715245			0.357623	+	0.933866i	$0.383587\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			3.67667i			0.289762i
$162$		0			0
$163$	−1.89972	−	1.09681i	−0.148798	−	0.0859085i	0.423753	−	0.905778i	$-0.360713\pi$
$163$	−1.89972	−	1.09681i	−0.148798	−	0.0859085i	−0.572550	+	0.819869i	$0.694046\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−12.9715	+	7.48911i	−1.00377	+	0.579525i	−0.909361	−	0.416009i	$-0.863428\pi$
$167$	−12.9715	+	7.48911i	−1.00377	+	0.579525i	−0.0944059	+	0.995534i	$0.530095\pi$
$168$		0			0
$169$	12.0343	+	4.91693i	0.925714	+	0.378225i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.99742	−	3.45963i	0.151861	−	0.263030i	−0.780051	−	0.625716i	$-0.784807\pi$
$173$	1.99742	−	3.45963i	0.151861	−	0.263030i	0.931912	+	0.362686i	$0.118140\pi$
$174$		0			0
$175$	1.76168	+	1.01711i	0.133171	+	0.0768862i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4.30591	+	7.45805i	0.321839	+	0.557441i	0.980867	−	0.194677i	$-0.0623658\pi$
$179$	4.30591	+	7.45805i	0.321839	+	0.557441i	−0.659029	+	0.752118i	$0.729033\pi$
$180$		0			0
$181$	15.6308			1.16183			0.580913	−	0.813966i	$-0.302696\pi$
$181$	15.6308			1.16183			0.580913	+	0.813966i	$0.302696\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−5.91288	−	10.2414i	−0.434724	−	0.752964i
$186$		0			0
$187$		−	6.23152i		−	0.455694i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	13.5280	−	23.4311i	0.978848	−	1.69541i	0.312246	−	0.950001i	$-0.398919\pi$
$191$	13.5280	−	23.4311i	0.978848	−	1.69541i	0.666602	−	0.745414i	$-0.267748\pi$
$192$		0			0
$193$	20.7426	−	11.9757i	1.49308	−	0.862032i	0.493115	−	0.869964i	$-0.335858\pi$
$193$	20.7426	−	11.9757i	1.49308	−	0.862032i	0.999969	+	0.00793192i	$0.00252484\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−8.95941	+	5.17272i	−0.638332	+	0.368541i	−0.783972	−	0.620797i	$-0.786809\pi$
$197$	−8.95941	+	5.17272i	−0.638332	+	0.368541i	0.145640	+	0.989338i	$0.453476\pi$
$198$		0			0
$199$	6.78642	−	11.7544i	0.481077	−	0.833250i	−0.518687	−	0.854964i	$-0.673579\pi$
$199$	6.78642	−	11.7544i	0.481077	−	0.833250i	0.999764	+	0.0217145i	$0.00691247\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			0.375118i			0.0263281i
$204$		0			0
$205$	−7.48100	−	12.9575i	−0.522496	−	0.904989i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.88092			0.130106
$210$		0			0
$211$	0.973098	+	1.68546i	0.0669909	+	0.116032i	0.897575	−	0.440861i	$-0.145327\pi$
$211$	0.973098	+	1.68546i	0.0669909	+	0.116032i	−0.830585	+	0.556893i	$0.811994\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−10.1676	−	5.87028i	−0.693427	−	0.400350i
$216$		0			0
$217$	0.0727592	−	0.126023i	0.00493922	−	0.00855498i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	13.1693	−	15.0899i	0.885860	−	1.01505i
$222$		0			0
$223$	−20.7156	+	11.9602i	−1.38722	+	0.800911i	−0.993001	−	0.118106i	$-0.962318\pi$
$223$	−20.7156	+	11.9602i	−1.38722	+	0.800911i	−0.394218	+	0.919017i	$0.628984\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	15.3084	+	8.83834i	1.01606	+	0.586621i	0.912959	−	0.408050i	$-0.133791\pi$
$227$	15.3084	+	8.83834i	1.01606	+	0.586621i	0.103098	+	0.994671i	$0.467125\pi$
$228$		0			0
$229$			10.9282i			0.722156i	0.932536	+	0.361078i	$0.117591\pi$
$229$			10.9282i			0.722156i	−0.932536	+	0.361078i	$0.882409\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	11.5970			0.759743			0.379871	−	0.925039i	$-0.375968\pi$
$233$	11.5970			0.759743			0.379871	+	0.925039i	$0.375968\pi$
$234$		0			0
$235$	28.1866			1.83869
$236$		0			0
$237$		0			0
$238$		0			0
$239$			18.0221i			1.16575i	0.812561	+	0.582877i	$0.198073\pi$
$239$			18.0221i			1.16575i	−0.812561	+	0.582877i	$0.801927\pi$
$240$		0			0
$241$	−0.709837	−	0.409825i	−0.0457246	−	0.0263991i	0.476963	−	0.878923i	$-0.341737\pi$
$241$	−0.709837	−	0.409825i	−0.0457246	−	0.0263991i	−0.522688	+	0.852524i	$0.675071\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	12.9789	−	7.49337i	0.829191	−	0.478734i
$246$		0			0
$247$	4.55471	+	3.97499i	0.289809	+	0.252923i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−7.17873	+	12.4339i	−0.453117	+	0.784822i	−0.998578	−	0.0533143i	$-0.983021\pi$
$251$	−7.17873	+	12.4339i	−0.453117	+	0.784822i	0.545460	+	0.838137i	$0.316355\pi$
$252$		0			0
$253$	3.18409	+	1.83834i	0.200182	+	0.115575i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	10.2985	+	17.8375i	0.642402	+	1.11267i	0.984895	+	0.173152i	$0.0553953\pi$
$257$	10.2985	+	17.8375i	0.642402	+	1.11267i	−0.342493	+	0.939520i	$0.611271\pi$
$258$		0			0
$259$	5.08243			0.315807
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−14.5115	−	25.1347i	−0.894820	−	1.54987i	−0.834028	−	0.551723i	$-0.813971\pi$
$263$	−14.5115	−	25.1347i	−0.894820	−	1.54987i	−0.0607920	−	0.998150i	$-0.519363\pi$
$264$		0			0
$265$		−	10.8265i		−	0.665066i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4.31381	+	7.47173i	−0.263017	+	0.455560i	−0.967042	−	0.254616i	$-0.918051\pi$
$269$	−4.31381	+	7.47173i	−0.263017	+	0.455560i	0.704025	+	0.710175i	$0.251384\pi$
$270$		0			0
$271$	−13.8459	+	7.99395i	−0.841080	+	0.485598i	−0.857631	−	0.514265i	$-0.828065\pi$
$271$	−13.8459	+	7.99395i	−0.841080	+	0.485598i	0.0165513	+	0.999863i	$0.494731\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.76168	−	1.01711i	0.106234	−	0.0613340i
$276$		0			0
$277$	−14.3728	+	24.8945i	−0.863579	+	1.49576i	0.00487180	+	0.999988i	$0.498449\pi$
$277$	−14.3728	+	24.8945i	−0.863579	+	1.49576i	−0.868451	+	0.495775i	$0.834884\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	5.83386i		−	0.348019i	−0.984744	−	0.174009i	$-0.944328\pi$
$281$		−	5.83386i		−	0.348019i	0.984744	−	0.174009i	$-0.0556723\pi$
$282$		0			0
$283$	13.8459	+	23.9818i	0.823055	+	1.42557i	0.903397	+	0.428805i	$0.141065\pi$
$283$	13.8459	+	23.9818i	0.823055	+	1.42557i	−0.0803425	+	0.996767i	$0.525601\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	6.43031			0.379569
$288$		0			0
$289$	−6.92820	−	12.0000i	−0.407541	−	0.705882i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	13.7980	+	7.96626i	0.806085	+	0.465394i	0.845595	−	0.533826i	$-0.179246\pi$
$293$	13.7980	+	7.96626i	0.806085	+	0.465394i	−0.0395092	+	0.999219i	$0.512579\pi$
$294$		0			0
$295$	−0.828165	+	1.43442i	−0.0482176	+	0.0835154i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.82539	+	11.1806i	0.221228	+	0.646592i
$300$		0			0
$301$	4.36980	−	2.52291i	0.251871	−	0.145418i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−10.6420	−	6.14415i	−0.609358	−	0.351813i
$306$		0			0
$307$			20.3955i			1.16403i	0.813178	+	0.582015i	$0.197736\pi$
$307$			20.3955i			1.16403i	−0.813178	+	0.582015i	$0.802264\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−12.9989			−0.737103			−0.368551	−	0.929607i	$-0.620146\pi$
$311$	−12.9989			−0.737103			−0.368551	+	0.929607i	$0.620146\pi$
$312$		0			0
$313$	−9.51274			−0.537692			−0.268846	−	0.963183i	$-0.586642\pi$
$313$	−9.51274			−0.537692			−0.268846	+	0.963183i	$0.586642\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	19.1354i		−	1.07475i	−0.843343	−	0.537376i	$-0.819416\pi$
$317$		−	19.1354i		−	1.07475i	0.843343	−	0.537376i	$-0.180584\pi$
$318$		0			0
$319$	0.324862	+	0.187559i	0.0181888	+	0.0105013i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	8.06587	−	4.65683i	0.448797	−	0.259113i
$324$		0			0
$325$	6.41547	+	1.26005i	0.355866	+	0.0698949i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6.05696	+	10.4910i	−0.333931	+	0.578385i
$330$		0			0
$331$	−9.78896	−	5.65166i	−0.538050	−	0.310643i	0.206238	−	0.978502i	$-0.433878\pi$
$331$	−9.78896	−	5.65166i	−0.538050	−	0.310643i	−0.744288	+	0.667858i	$0.767211\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−18.7431	−	32.4640i	−1.02405	−	1.77370i
$336$		0			0
$337$	−5.14035			−0.280013			−0.140006	−	0.990151i	$-0.544712\pi$
$337$	−5.14035			−0.280013			−0.140006	+	0.990151i	$0.544712\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−0.0727592	−	0.126023i	−0.00394013	−	0.00682451i
$342$		0			0
$343$			14.2937i			0.771785i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−4.36129	+	7.55397i	−0.234126	+	0.405518i	−0.959018	−	0.283344i	$-0.908556\pi$
$347$	−4.36129	+	7.55397i	−0.234126	+	0.405518i	0.724892	+	0.688862i	$0.241890\pi$
$348$		0			0
$349$	23.4625	−	13.5461i	1.25592	−	0.725104i	0.283639	−	0.958931i	$-0.408458\pi$
$349$	23.4625	−	13.5461i	1.25592	−	0.725104i	0.972278	+	0.233827i	$0.0751248\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	19.9524	−	11.5195i	1.06196	−	0.613123i	0.135986	−	0.990711i	$-0.456580\pi$
$353$	19.9524	−	11.5195i	1.06196	−	0.613123i	0.925974	+	0.377588i	$0.123246\pi$
$354$		0			0
$355$	−11.2304	+	19.4516i	−0.596046	+	1.03238i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			1.13392i			0.0598462i	0.999552	+	0.0299231i	$0.00952624\pi$
$359$			1.13392i			0.0598462i	−0.999552	+	0.0299231i	$0.990474\pi$
$360$		0			0
$361$	−8.09439	−	14.0199i	−0.426020	−	0.737889i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−25.9578			−1.35869
$366$		0			0
$367$	−14.3992	−	24.9401i	−0.751632	−	1.30186i	−0.947031	−	0.321141i	$-0.895934\pi$
$367$	−14.3992	−	24.9401i	−0.751632	−	1.30186i	0.195399	−	0.980724i	$-0.437400\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	4.02959	+	2.32648i	0.209206	+	0.120785i
$372$		0			0
$373$	11.8939	−	20.6008i	0.615842	−	1.06667i	−0.374394	−	0.927270i	$-0.622149\pi$
$373$	11.8939	−	20.6008i	0.615842	−	1.06667i	0.990236	−	0.139400i	$-0.0445173\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	0.390291	+	1.14072i	0.0201010	+	0.0587501i
$378$		0			0
$379$	−12.9434	+	7.47289i	−0.664859	+	0.383856i	−0.794126	−	0.607753i	$-0.792071\pi$
$379$	−12.9434	+	7.47289i	−0.664859	+	0.383856i	0.129267	+	0.991610i	$0.458738\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	3.92940	+	2.26864i	0.200783	+	0.115922i	0.597021	−	0.802226i	$-0.296351\pi$
$383$	3.92940	+	2.26864i	0.200783	+	0.115922i	−0.396238	+	0.918148i	$0.629684\pi$
$384$		0			0
$385$			3.28491i			0.167414i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	9.51685			0.482524			0.241262	−	0.970460i	$-0.422439\pi$
$389$	9.51685			0.482524			0.241262	+	0.970460i	$0.422439\pi$
$390$		0			0
$391$	18.2056			0.920698
$392$		0			0
$393$		0			0
$394$		0			0
$395$			35.4913i			1.78576i
$396$		0			0
$397$	−29.8307	−	17.2227i	−1.49716	−	0.864385i	−0.497163	−	0.867657i	$-0.665625\pi$
$397$	−29.8307	−	17.2227i	−1.49716	−	0.864385i	−0.999995	+	0.00327253i	$0.998958\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	12.2605	−	7.07859i	0.612259	−	0.353488i	−0.161590	−	0.986858i	$-0.551662\pi$
$401$	12.2605	−	7.07859i	0.612259	−	0.353488i	0.773849	+	0.633370i	$0.218329\pi$
$402$		0			0
$403$	0.0901380	−	0.458933i	0.00449009	−	0.0228611i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	2.54121	−	4.40151i	0.125963	−	0.218175i
$408$		0			0
$409$	−0.299539	−	0.172939i	−0.0148112	−	0.00855127i	0.492576	−	0.870269i	$-0.336055\pi$
$409$	−0.299539	−	0.172939i	−0.0148112	−	0.00855127i	−0.507387	+	0.861718i	$0.669389\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−0.355925	−	0.616481i	−0.0175139	−	0.0303350i
$414$		0			0
$415$	−15.0255			−0.737571
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−6.98890	−	12.1051i	−0.341430	−	0.591374i	0.643268	−	0.765641i	$-0.277578\pi$
$419$	−6.98890	−	12.1051i	−0.341430	−	0.591374i	−0.984699	+	0.174266i	$0.944245\pi$
$420$		0			0
$421$			22.1609i			1.08006i	0.841647	+	0.540028i	$0.181586\pi$
$421$			22.1609i			1.08006i	−0.841647	+	0.540028i	$0.818414\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	5.03638	−	8.72327i	0.244300	−	0.423141i
$426$		0			0
$427$	4.57366	−	2.64061i	0.221335	−	0.127788i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−22.2109	+	12.8235i	−1.06986	+	0.617686i	−0.928144	−	0.372221i	$-0.878596\pi$
$431$	−22.2109	+	12.8235i	−1.06986	+	0.617686i	−0.141719	+	0.989907i	$0.545263\pi$
$432$		0			0
$433$	−5.07802	+	8.79538i	−0.244034	+	0.422679i	−0.961860	−	0.273544i	$-0.911804\pi$
$433$	−5.07802	+	8.79538i	−0.244034	+	0.422679i	0.717826	+	0.696223i	$0.245137\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			5.49516i			0.262869i
$438$		0			0
$439$	−10.6982	−	18.5298i	−0.510598	−	0.884381i	−0.999925	−	0.0122808i	$-0.996091\pi$
$439$	−10.6982	−	18.5298i	−0.510598	−	0.884381i	0.489327	−	0.872100i	$-0.337243\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−10.6688			−0.506889			−0.253444	−	0.967350i	$-0.581563\pi$
$443$	−10.6688			−0.506889			−0.253444	+	0.967350i	$0.581563\pi$
$444$		0			0
$445$	−18.7225	−	32.4283i	−0.887531	−	1.53725i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	8.62407	+	4.97911i	0.406995	+	0.234979i	0.689498	−	0.724288i	$-0.257831\pi$
$449$	8.62407	+	4.97911i	0.406995	+	0.234979i	−0.282503	+	0.959266i	$0.591165\pi$
$450$		0			0
$451$	3.21515	−	5.56881i	0.151396	−	0.262225i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−6.94208	+	7.95452i	−0.325450	+	0.372914i
$456$		0			0
$457$	−23.2035	+	13.3966i	−1.08542	+	0.626665i	−0.932352	−	0.361552i	$-0.882247\pi$
$457$	−23.2035	+	13.3966i	−1.08542	+	0.626665i	−0.153063	+	0.988216i	$0.548914\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−27.7410	−	16.0163i	−1.29203	−	0.745952i	−0.313014	−	0.949749i	$-0.601339\pi$
$461$	−27.7410	−	16.0163i	−1.29203	−	0.745952i	−0.979013	+	0.203796i	$0.934672\pi$
$462$		0			0
$463$			21.7564i			1.01110i	0.862796	+	0.505552i	$0.168711\pi$
$463$			21.7564i			1.01110i	−0.862796	+	0.505552i	$0.831289\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−33.7322			−1.56094			−0.780469	−	0.625195i	$-0.785020\pi$
$467$	−33.7322			−1.56094			−0.780469	+	0.625195i	$0.785020\pi$
$468$		0			0
$469$	16.1107			0.743922
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	5.04581i		−	0.232007i
$474$		0			0
$475$	2.63302	+	1.52017i	0.120811	+	0.0697504i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	9.17041	−	5.29454i	0.419007	−	0.241914i	−0.275646	−	0.961259i	$-0.588892\pi$
$479$	9.17041	−	5.29454i	0.419007	−	0.241914i	0.694652	+	0.719346i	$0.255558\pi$
$480$		0			0
$481$	15.4555	−	5.28801i	0.704709	−	0.241112i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−11.7564	+	20.3626i	−0.533829	+	0.924619i
$486$		0			0
$487$	−19.0612	−	11.0050i	−0.863746	−	0.498684i	0.00151867	−	0.999999i	$-0.499517\pi$
$487$	−19.0612	−	11.0050i	−0.863746	−	0.498684i	−0.865265	+	0.501315i	$0.832850\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	8.25727	+	14.3020i	0.372645	+	0.645441i	0.989972	−	0.141266i	$-0.0451174\pi$
$491$	8.25727	+	14.3020i	0.372645	+	0.645441i	−0.617326	+	0.786707i	$0.711784\pi$
$492$		0			0
$493$	1.85746			0.0836557
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4.82654	−	8.35982i	−0.216500	−	0.374989i
$498$		0			0
$499$			11.1236i			0.497960i	0.968509	+	0.248980i	$0.0800954\pi$
$499$			11.1236i			0.497960i	−0.968509	+	0.248980i	$0.919905\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−14.6361	+	25.3504i	−0.652591	+	1.13032i	0.329901	+	0.944015i	$0.392985\pi$
$503$	−14.6361	+	25.3504i	−0.652591	+	1.13032i	−0.982492	+	0.186305i	$0.940349\pi$
$504$		0			0
$505$	−6.53121	+	3.77080i	−0.290635	+	0.167798i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−24.4571	+	14.1203i	−1.08404	+	0.625871i	−0.931983	−	0.362501i	$-0.881923\pi$
$509$	−24.4571	+	14.1203i	−1.08404	+	0.625871i	−0.152057	+	0.988372i	$0.548590\pi$
$510$		0			0
$511$	5.57802	−	9.66141i	0.246757	−	0.427396i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			13.6267i			0.600462i
$516$		0			0
$517$	6.05696	+	10.4910i	0.266385	+	0.461392i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	18.6401			0.816636			0.408318	−	0.912840i	$-0.366116\pi$
$521$	18.6401			0.816636			0.408318	+	0.912840i	$0.366116\pi$
$522$		0			0
$523$	−17.6561	−	30.5812i	−0.772047	−	1.33722i	−0.936439	−	0.350830i	$-0.885900\pi$
$523$	−17.6561	−	30.5812i	−0.772047	−	1.33722i	0.164392	−	0.986395i	$-0.447434\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−0.624022	−	0.360279i	−0.0271828	−	0.0156940i
$528$		0			0
$529$	6.12924	−	10.6161i	0.266489	−	0.461572i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	19.5543	−	6.69041i	0.846992	−	0.289794i
$534$		0			0
$535$	−13.1840	+	7.61181i	−0.569996	+	0.329087i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	5.57802	+	3.22047i	0.240262	+	0.138715i
$540$		0			0
$541$			45.2196i			1.94414i	0.234682	+	0.972072i	$0.424595\pi$
$541$			45.2196i			1.94414i	−0.234682	+	0.972072i	$0.575405\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	7.64330			0.327403
$546$		0			0
$547$	−37.8808			−1.61967			−0.809834	−	0.586660i	$-0.800443\pi$
$547$	−37.8808			−1.61967			−0.809834	+	0.586660i	$0.800443\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			0.560653i			0.0238846i
$552$		0			0
$553$	−13.2097	−	7.62665i	−0.561736	−	0.324318i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	17.8759	−	10.3206i	0.757426	−	0.437300i	−0.0709450	−	0.997480i	$-0.522601\pi$
$557$	17.8759	−	10.3206i	0.757426	−	0.437300i	0.828371	+	0.560180i	$0.189268\pi$
$558$		0			0
$559$	10.6635	−	12.2186i	0.451016	−	0.516793i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	10.2300	−	17.7189i	0.431143	−	0.746761i	−0.565829	−	0.824522i	$-0.691444\pi$
$563$	10.2300	−	17.7189i	0.431143	−	0.746761i	0.996972	+	0.0777613i	$0.0247772\pi$
$564$		0			0
$565$	11.0451	+	6.37691i	0.464673	+	0.268279i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	10.5681	+	18.3044i	0.443037	+	0.767362i	0.997913	−	0.0645708i	$-0.0205678\pi$
$569$	10.5681	+	18.3044i	0.443037	+	0.767362i	−0.554877	+	0.831933i	$0.687235\pi$
$570$		0			0
$571$	−13.9221			−0.582621			−0.291310	−	0.956629i	$-0.594091\pi$
$571$	−13.9221			−0.582621			−0.291310	+	0.956629i	$0.594091\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.97152	+	5.14683i	0.123921	+	0.214637i
$576$		0			0
$577$			26.2420i			1.09247i	0.837633	+	0.546234i	$0.183939\pi$
$577$			26.2420i			1.09247i	−0.837633	+	0.546234i	$0.816061\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3.22879	−	5.59243i	0.133953	−	0.232013i
$582$		0			0
$583$	4.02959	−	2.32648i	0.166889	−	0.0963531i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−21.3670	+	12.3362i	−0.881910	+	0.509171i	−0.871288	−	0.490773i	$-0.836715\pi$
$587$	−21.3670	+	12.3362i	−0.881910	+	0.509171i	−0.0106221	+	0.999944i	$0.503381\pi$
$588$		0			0
$589$	0.108746	−	0.188354i	0.00448081	−	0.00776099i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			40.3506i			1.65700i	0.559988	+	0.828501i	$0.310806\pi$
$593$			40.3506i			1.65700i	−0.559988	+	0.828501i	$0.689194\pi$
$594$		0			0
$595$	8.13287	+	14.0865i	0.333415	+	0.577492i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	4.96104			0.202702			0.101351	−	0.994851i	$-0.467683\pi$
$599$	4.96104			0.202702			0.101351	+	0.994851i	$0.467683\pi$
$600$		0			0
$601$	7.73846	+	13.4034i	0.315658	+	0.546737i	0.979577	−	0.201068i	$-0.0644413\pi$
$601$	7.73846	+	13.4034i	0.315658	+	0.546737i	−0.663919	+	0.747805i	$0.731108\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−22.0210	−	12.7138i	−0.895282	−	0.516891i
$606$		0			0
$607$	−11.2167	+	19.4279i	−0.455273	+	0.788556i	−0.998704	−	0.0508979i	$-0.983792\pi$
$607$	−11.2167	+	19.4279i	−0.455273	+	0.788556i	0.543431	+	0.839454i	$0.317125\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−7.50368	+	38.2046i	−0.303566	+	1.54559i
$612$		0			0
$613$	14.3634	−	8.29274i	0.580134	−	0.334941i	−0.181053	−	0.983473i	$-0.557950\pi$
$613$	14.3634	−	8.29274i	0.580134	−	0.334941i	0.761187	+	0.648533i	$0.224617\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−28.6107	−	16.5184i	−1.15182	−	0.665005i	−0.202492	−	0.979284i	$-0.564904\pi$
$617$	−28.6107	−	16.5184i	−1.15182	−	0.665005i	−0.949331	+	0.314279i	$0.898237\pi$
$618$		0			0
$619$		−	35.3913i		−	1.42249i	−0.702942	−	0.711247i	$-0.748131\pi$
$619$		−	35.3913i		−	1.42249i	0.702942	−	0.711247i	$-0.251869\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	16.0929			0.644750
$624$		0			0
$625$	−30.7785			−1.23114
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	25.1665i		−	1.00345i
$630$		0			0
$631$	−5.58850	−	3.22652i	−0.222475	−	0.128446i	0.384621	−	0.923075i	$-0.374332\pi$
$631$	−5.58850	−	3.22652i	−0.222475	−	0.128446i	−0.607096	+	0.794629i	$0.707666\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	23.3276	−	13.4682i	0.925729	−	0.534470i
$636$		0			0
$637$	6.70147	+	19.5867i	0.265522	+	0.776052i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	10.8871	−	18.8571i	0.430016	−	0.744810i	−0.566858	−	0.823815i	$-0.691841\pi$
$641$	10.8871	−	18.8571i	0.430016	−	0.744810i	0.996874	+	0.0790058i	$0.0251746\pi$
$642$		0			0
$643$	34.8552	+	20.1237i	1.37456	+	0.793600i	0.991498	−	0.130125i	$-0.0415378\pi$
$643$	34.8552	+	20.1237i	1.37456	+	0.793600i	0.383058	+	0.923724i	$0.374871\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−18.8105	−	32.5808i	−0.739519	−	1.28088i	−0.952712	−	0.303874i	$-0.901720\pi$
$647$	−18.8105	−	32.5808i	−0.739519	−	1.28088i	0.213193	−	0.977010i	$-0.431614\pi$
$648$		0			0
$649$	−0.711850			−0.0279426
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−15.3559	−	26.5972i	−0.600924	−	1.04083i	−0.992681	−	0.120762i	$-0.961466\pi$
$653$	−15.3559	−	26.5972i	−0.600924	−	1.04083i	0.391758	−	0.920068i	$-0.371867\pi$
$654$		0			0
$655$			0.289112i			0.0112965i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	6.88489	−	11.9250i	0.268197	−	0.464531i	−0.700199	−	0.713947i	$-0.746906\pi$
$659$	6.88489	−	11.9250i	0.268197	−	0.464531i	0.968396	+	0.249417i	$0.0802388\pi$
$660$		0			0
$661$	−8.85492	+	5.11239i	−0.344416	+	0.198849i	−0.662223	−	0.749307i	$-0.730387\pi$
$661$	−8.85492	+	5.11239i	−0.344416	+	0.198849i	0.317807	+	0.948155i	$0.397054\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−4.25187	+	2.45482i	−0.164880	+	0.0951937i
$666$		0			0
$667$	−0.547961	+	0.949096i	−0.0212171	+	0.0367491i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	5.28121i		−	0.203879i
$672$		0			0
$673$	2.13077	+	3.69060i	0.0821351	+	0.142262i	0.904167	−	0.427180i	$-0.140493\pi$
$673$	2.13077	+	3.69060i	0.0821351	+	0.142262i	−0.822032	+	0.569442i	$0.807159\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	4.47885			0.172136			0.0860681	−	0.996289i	$-0.472570\pi$
$677$	4.47885			0.172136			0.0860681	+	0.996289i	$0.472570\pi$
$678$		0			0
$679$	−5.05260	−	8.75137i	−0.193901	−	0.335847i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−17.4014	−	10.0467i	−0.665846	−	0.384426i	0.128655	−	0.991689i	$-0.458934\pi$
$683$	−17.4014	−	10.0467i	−0.665846	−	0.384426i	−0.794501	+	0.607263i	$0.792267\pi$
$684$		0			0
$685$	9.07386	−	15.7164i	0.346694	−	0.600492i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	14.6744	+	2.88217i	0.559051	+	0.109802i
$690$		0			0
$691$	−32.6571	+	18.8546i	−1.24233	+	0.717261i	−0.969569	−	0.244820i	$-0.921271\pi$
$691$	−32.6571	+	18.8546i	−1.24233	+	0.717261i	−0.272764	+	0.962081i	$0.587938\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	23.5780	+	13.6128i	0.894365	+	0.516362i
$696$		0			0
$697$		−	31.8407i		−	1.20605i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	17.2849			0.652842			0.326421	−	0.945225i	$-0.394157\pi$
$701$	17.2849			0.652842			0.326421	+	0.945225i	$0.394157\pi$
$702$		0			0
$703$	7.59622			0.286497
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	3.24119i		−	0.121898i
$708$		0			0
$709$	15.7410	+	9.08807i	0.591166	+	0.341310i	0.765558	−	0.643366i	$-0.222463\pi$
$709$	15.7410	+	9.08807i	0.591166	+	0.341310i	−0.174393	+	0.984676i	$0.555796\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.368180	−	0.212569i	0.0137884	−	0.00796076i
$714$		0			0
$715$	3.41778	+	9.98928i	0.127818	+	0.373578i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	9.95668	−	17.2455i	0.371322	−	0.643148i	−0.618448	−	0.785826i	$-0.712238\pi$
$719$	9.95668	−	17.2455i	0.371322	−	0.643148i	0.989769	+	0.142678i	$0.0455714\pi$
$720$		0			0
$721$	−5.07180	−	2.92820i	−0.188884	−	0.109052i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	0.303174	+	0.525113i	0.0112596	+	0.0195022i
$726$		0			0
$727$	38.7446			1.43696			0.718479	−	0.695549i	$-0.244839\pi$
$727$	38.7446			1.43696			0.718479	+	0.695549i	$0.244839\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−12.4926	−	21.6378i	−0.462055	−	0.800302i
$732$		0			0
$733$		−	3.19477i		−	0.118001i	−0.998258	−	0.0590007i	$-0.981209\pi$
$733$		−	3.19477i		−	0.118001i	0.998258	−	0.0590007i	$-0.0187914\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	8.05534	−	13.9523i	0.296722	−	0.513938i
$738$		0			0
$739$	35.2234	−	20.3362i	1.29571	−	0.748080i	0.316053	−	0.948742i	$-0.397642\pi$
$739$	35.2234	−	20.3362i	1.29571	−	0.748080i	0.979661	+	0.200661i	$0.0643091\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	12.0464	−	6.95498i	0.441939	−	0.255154i	−0.262481	−	0.964937i	$-0.584541\pi$
$743$	12.0464	−	6.95498i	0.441939	−	0.255154i	0.704420	+	0.709784i	$0.251207\pi$
$744$		0			0
$745$	−12.5322	+	21.7064i	−0.459143	+	0.795260i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	6.54275i		−	0.239067i
$750$		0			0
$751$	−8.79317	−	15.2302i	−0.320867	−	0.555759i	0.659800	−	0.751441i	$-0.270641\pi$
$751$	−8.79317	−	15.2302i	−0.320867	−	0.555759i	−0.980667	+	0.195683i	$0.937308\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	35.8299			1.30398
$756$		0			0
$757$	−11.0553	−	19.1484i	−0.401813	−	0.695961i	0.592132	−	0.805841i	$-0.298286\pi$
$757$	−11.0553	−	19.1484i	−0.401813	−	0.695961i	−0.993945	+	0.109881i	$0.964953\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−21.0765	−	12.1685i	−0.764022	−	0.441108i	0.0667160	−	0.997772i	$-0.478748\pi$
$761$	−21.0765	−	12.1685i	−0.764022	−	0.441108i	−0.830738	+	0.556664i	$0.812081\pi$
$762$		0			0
$763$	−1.64245	+	2.84481i	−0.0594608	+	0.102989i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.72377	−	1.50437i	−0.0622418	−	0.0543198i
$768$		0			0
$769$	6.14197	−	3.54607i	0.221485	−	0.127875i	−0.385153	−	0.922853i	$-0.625851\pi$
$769$	6.14197	−	3.54607i	0.221485	−	0.127875i	0.606638	+	0.794978i	$0.292518\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−0.840035	−	0.484994i	−0.0302140	−	0.0174440i	0.484817	−	0.874616i	$-0.338886\pi$
$773$	−0.840035	−	0.484994i	−0.0302140	−	0.0174440i	−0.515031	+	0.857172i	$0.672220\pi$
$774$		0			0
$775$		−	0.235219i		−	0.00844931i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	9.61076			0.344341
$780$		0			0
$781$	−9.65309			−0.345415
$782$		0			0
$783$		0			0
$784$		0			0
$785$			23.3929i			0.834929i
$786$		0			0
$787$	−0.0361875	−	0.0208929i	−0.00128994	−	0.000744750i	0.499355	−	0.866398i	$-0.333570\pi$
$787$	−0.0361875	−	0.0208929i	−0.00128994	−	0.000744750i	−0.500645	+	0.865653i	$0.666904\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−4.74693	+	2.74064i	−0.168782	+	0.0974461i
$792$		0			0
$793$	11.1609	−	12.7886i	0.396336	−	0.454138i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−19.6671	+	34.0645i	−0.696646	+	1.20663i	0.272977	+	0.962021i	$0.411992\pi$
$797$	−19.6671	+	34.0645i	−0.696646	+	1.20663i	−0.969623	+	0.244606i	$0.921341\pi$
$798$		0			0
$799$	51.9477	+	29.9920i	1.83778	+	1.06104i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−5.57802	−	9.66141i	−0.196844	−	0.340944i
$804$		0			0
$805$	−9.59697			−0.338249
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.404078	−	0.699884i	−0.0142066	−	0.0246066i	0.858835	−	0.512253i	$-0.171189\pi$
$809$	−0.404078	−	0.699884i	−0.0142066	−	0.0246066i	−0.873041	+	0.487646i	$0.837856\pi$
$810$		0			0
$811$			24.3955i			0.856640i	0.903627	+	0.428320i	$0.140894\pi$
$811$			24.3955i			0.856640i	−0.903627	+	0.428320i	$0.859106\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2.86292	−	4.95873i	0.100284	−	0.173697i
$816$		0			0
$817$	6.53112	−	3.77075i	0.228495	−	0.131922i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	15.0406	−	8.68369i	0.524920	−	0.303063i	−0.214025	−	0.976828i	$-0.568657\pi$
$821$	15.0406	−	8.68369i	0.524920	−	0.303063i	0.738945	+	0.673765i	$0.235324\pi$
$822$		0			0
$823$	2.83865	−	4.91669i	0.0989491	−	0.171385i	−0.812301	−	0.583239i	$-0.801785\pi$
$823$	2.83865	−	4.91669i	0.0989491	−	0.171385i	0.911250	+	0.411854i	$0.135119\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			14.5094i			0.504540i	0.967657	+	0.252270i	$0.0811772\pi$
$827$			14.5094i			0.504540i	−0.967657	+	0.252270i	$0.918823\pi$
$828$		0			0
$829$	23.7516	+	41.1390i	0.824928	+	1.42882i	0.901974	+	0.431790i	$0.142118\pi$
$829$	23.7516	+	41.1390i	0.824928	+	1.42882i	−0.0770456	+	0.997028i	$0.524549\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	31.8934			1.10504
$834$		0			0
$835$	−19.5483	−	33.8587i	−0.676498	−	1.17173i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1.78192	−	1.02879i	−0.0615188	−	0.0355179i	0.468925	−	0.883238i	$-0.344641\pi$
$839$	−1.78192	−	1.02879i	−0.0615188	−	0.0355179i	−0.530444	+	0.847720i	$0.677975\pi$
$840$		0			0
$841$	14.4441	−	25.0179i	0.498072	−	0.862686i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−12.8343	+	31.4123i	−0.441515	+	1.08062i
$846$		0			0
$847$	9.46410	−	5.46410i	0.325190	−	0.187749i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	12.8592	+	7.42425i	0.440807	+	0.254500i
$852$		0			0
$853$		−	4.10363i		−	0.140506i	−0.997529	−	0.0702528i	$-0.977619\pi$
$853$		−	4.10363i		−	0.140506i	0.997529	−	0.0702528i	$-0.0223806\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−35.9776			−1.22897			−0.614486	−	0.788928i	$-0.710636\pi$
$857$	−35.9776			−1.22897			−0.614486	+	0.788928i	$0.710636\pi$
$858$		0			0
$859$	12.7711			0.435745			0.217872	−	0.975977i	$-0.430088\pi$
$859$	12.7711			0.435745			0.217872	+	0.975977i	$0.430088\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	34.2815i		−	1.16696i	−0.812128	−	0.583479i	$-0.801691\pi$
$863$		−	34.2815i		−	1.16696i	0.812128	−	0.583479i	$-0.198309\pi$
$864$		0			0
$865$	9.03043	+	5.21372i	0.307044	+	0.177272i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−13.2097	+	7.62665i	−0.448110	+	0.258716i
$870$		0			0
$871$	48.9920	−	16.7623i	1.66003	−	0.567970i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	4.66562	−	8.08108i	0.157727	−	0.273190i
$876$		0			0
$877$	−39.4625	−	22.7837i	−1.33255	−	0.769351i	−0.346864	−	0.937915i	$-0.612753\pi$
$877$	−39.4625	−	22.7837i	−1.33255	−	0.769351i	−0.985691	+	0.168565i	$0.946087\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	6.88302	+	11.9217i	0.231895	+	0.401654i	0.958366	−	0.285544i	$-0.0921742\pi$
$881$	6.88302	+	11.9217i	0.231895	+	0.401654i	−0.726471	+	0.687197i	$0.758841\pi$
$882$		0			0
$883$	−1.44199			−0.0485269			−0.0242634	−	0.999706i	$-0.507724\pi$
$883$	−1.44199			−0.0485269			−0.0242634	+	0.999706i	$0.507724\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	13.2859	+	23.0118i	0.446097	+	0.772662i	0.998128	−	0.0611612i	$-0.0194804\pi$
$887$	13.2859	+	23.0118i	0.446097	+	0.772662i	−0.552031	+	0.833824i	$0.686147\pi$
$888$		0			0
$889$			11.5766i			0.388268i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−9.05275	+	15.6798i	−0.302939	+	0.524705i
$894$		0			0
$895$	−19.4673	+	11.2394i	−0.650719	+	0.375693i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	0.0375641	−	0.0216877i	0.00125283	−	0.000723324i
$900$		0			0
$901$	11.5200	−	19.9532i	0.383786	−	0.664736i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			40.8000i			1.35624i
$906$		0			0
$907$	−5.03911	−	8.72800i	−0.167321	−	0.289808i	0.770156	−	0.637855i	$-0.220178\pi$
$907$	−5.03911	−	8.72800i	−0.167321	−	0.289808i	−0.937477	+	0.348047i	$0.886845\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	44.5252			1.47518			0.737592	−	0.675246i	$-0.235963\pi$
$911$	44.5252			1.47518			0.737592	+	0.675246i	$0.235963\pi$
$912$		0			0
$913$	−3.22879	−	5.59243i	−0.106857	−	0.185083i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−0.107607	−	0.0621267i	−0.00355348	−	0.00205160i
$918$		0			0
$919$	16.7591	−	29.0275i	0.552830	−	0.957530i	−0.445239	−	0.895412i	$-0.646881\pi$
$919$	16.7591	−	29.0275i	0.552830	−	0.957530i	0.998069	−	0.0621180i	$-0.0197855\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−23.3753	−	20.4001i	−0.769407	−	0.671478i
$924$		0			0
$925$	7.11469	−	4.10767i	0.233930	−	0.135059i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−15.1077	−	8.72243i	−0.495667	−	0.286174i	0.231255	−	0.972893i	$-0.425717\pi$
$929$	−15.1077	−	8.72243i	−0.495667	−	0.286174i	−0.726923	+	0.686719i	$0.759050\pi$
$930$		0			0
$931$			9.62665i			0.315501i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	16.2657			0.531947
$936$		0			0
$937$	17.0052			0.555535			0.277767	−	0.960648i	$-0.410406\pi$
$937$	17.0052			0.555535			0.277767	+	0.960648i	$0.410406\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			15.7406i			0.513128i	0.966527	+	0.256564i	$0.0825904\pi$
$941$			15.7406i			0.513128i	−0.966527	+	0.256564i	$0.917410\pi$
$942$		0			0
$943$	16.2695	+	9.39319i	0.529807	+	0.305884i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	28.2102	−	16.2872i	0.916709	−	0.529262i	0.0341252	−	0.999418i	$-0.489136\pi$
$947$	28.2102	−	16.2872i	0.916709	−	0.529262i	0.882584	+	0.470155i	$0.155802\pi$
$948$		0			0
$949$	6.91034	−	35.1836i	0.224319	−	1.14211i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−15.6154	+	27.0466i	−0.505831	+	0.876125i	0.494146	+	0.869379i	$0.335481\pi$
$953$	−15.6154	+	27.0466i	−0.505831	+	0.876125i	−0.999977	+	0.00674630i	$0.997853\pi$
$954$		0			0
$955$	61.1607	+	35.3111i	1.97911	+	1.14264i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	3.89972	+	6.75452i	0.125929	+	0.218115i
$960$		0			0
$961$	30.9832			0.999457
$962$		0			0
$963$		0			0
$964$		0			0
$965$	31.2595	+	54.1430i	1.00628	+	1.74292i
$966$		0			0
$967$			37.8998i			1.21877i	0.792873	+	0.609387i	$0.208585\pi$
$967$			37.8998i			1.21877i	−0.792873	+	0.609387i	$0.791415\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−13.6628	+	23.6647i	−0.438461	+	0.759437i	−0.997571	−	0.0696563i	$-0.977810\pi$
$971$	−13.6628	+	23.6647i	−0.438461	+	0.759437i	0.559110	+	0.829094i	$0.311143\pi$
$972$		0			0
$973$	−10.1333	+	5.85044i	−0.324858	+	0.187557i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−25.8089	+	14.9008i	−0.825700	+	0.476718i	−0.852378	−	0.522926i	$-0.824840\pi$
$977$	−25.8089	+	14.9008i	−0.825700	+	0.476718i	0.0266779	+	0.999644i	$0.491507\pi$
$978$		0			0
$979$	8.04647	−	13.9369i	0.257166	−	0.445425i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	16.0704i		−	0.512565i	−0.966602	−	0.256282i	$-0.917502\pi$
$983$		−	16.0704i		−	0.512565i	0.966602	−	0.256282i	$-0.0824977\pi$
$984$		0			0
$985$	−13.5020	−	23.3862i	−0.430210	−	0.745145i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	14.7415			0.468753
$990$		0			0
$991$	−15.9772	−	27.6733i	−0.507533	−	0.879073i	−0.999962	−	0.00872027i	$-0.997224\pi$
$991$	−15.9772	−	27.6733i	−0.507533	−	0.879073i	0.492429	−	0.870353i	$-0.336109\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	30.6818	+	17.7142i	0.972679	+	0.561577i
$996$		0			0
$997$	4.39599	−	7.61407i	0.139222	−	0.241140i	−0.787980	−	0.615701i	$-0.788873\pi$
$997$	4.39599	−	7.61407i	0.139222	−	0.241140i	0.927202	+	0.374561i	$0.122206\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	936.2.bi.b.361.3		8
3.2	odd	2		104.2.o.a.49.4	yes	8
4.3	odd	2		1872.2.by.n.1297.3		8
12.11	even	2		208.2.w.c.49.1		8
13.4	even	6	inner	936.2.bi.b.433.2		8
24.5	odd	2		832.2.w.g.257.1		8
24.11	even	2		832.2.w.i.257.4		8
39.2	even	12		1352.2.a.k.1.1		4
39.5	even	4		1352.2.i.k.1329.4		8
39.8	even	4		1352.2.i.l.1329.4		8
39.11	even	12		1352.2.a.l.1.1		4
39.17	odd	6		104.2.o.a.17.4	✓	8
39.20	even	12		1352.2.i.l.529.4		8
39.23	odd	6		1352.2.f.f.337.2		8
39.29	odd	6		1352.2.f.f.337.1		8
39.32	even	12		1352.2.i.k.529.4		8
39.35	odd	6		1352.2.o.f.1161.4		8
39.38	odd	2		1352.2.o.f.361.4		8
52.43	odd	6		1872.2.by.n.433.2		8
156.11	odd	12		2704.2.a.be.1.4		4
156.23	even	6		2704.2.f.q.337.8		8
156.95	even	6		208.2.w.c.17.1		8
156.107	even	6		2704.2.f.q.337.7		8
156.119	odd	12		2704.2.a.bd.1.4		4
312.173	odd	6		832.2.w.g.641.1		8
312.251	even	6		832.2.w.i.641.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.o.a.17.4	✓	8	39.17	odd	6
104.2.o.a.49.4	yes	8	3.2	odd	2
208.2.w.c.17.1		8	156.95	even	6
208.2.w.c.49.1		8	12.11	even	2
832.2.w.g.257.1		8	24.5	odd	2
832.2.w.g.641.1		8	312.173	odd	6
832.2.w.i.257.4		8	24.11	even	2
832.2.w.i.641.4		8	312.251	even	6
936.2.bi.b.361.3		8	1.1	even	1	trivial
936.2.bi.b.433.2		8	13.4	even	6	inner
1352.2.a.k.1.1		4	39.2	even	12
1352.2.a.l.1.1		4	39.11	even	12
1352.2.f.f.337.1		8	39.29	odd	6
1352.2.f.f.337.2		8	39.23	odd	6
1352.2.i.k.529.4		8	39.32	even	12
1352.2.i.k.1329.4		8	39.5	even	4
1352.2.i.l.529.4		8	39.20	even	12
1352.2.i.l.1329.4		8	39.8	even	4
1352.2.o.f.361.4		8	39.38	odd	2
1352.2.o.f.1161.4		8	39.35	odd	6
1872.2.by.n.433.2		8	52.43	odd	6
1872.2.by.n.1297.3		8	4.3	odd	2
2704.2.a.bd.1.4		4	156.119	odd	12
2704.2.a.be.1.4		4	156.11	odd	12
2704.2.f.q.337.7		8	156.107	even	6
2704.2.f.q.337.8		8	156.23	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	936.bi (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+2.61023i q^{5} +(-0.971521 - 0.560908i) q^{7} +(-0.971521 + 0.560908i) q^{11} +(-3.53796 - 0.694883i) q^{13} +(-2.77743 + 4.81064i) q^{17} +(-1.45204 - 0.838335i) q^{19} +(-1.63871 - 2.83834i) q^{23} -1.81333 q^{25} +(-0.167192 - 0.289586i) q^{29} +0.129717i q^{31} +(1.46410 - 2.53590i) q^{35} +(-3.92356 + 2.26527i) q^{37} +(-4.96410 + 2.86603i) q^{41} +(-2.24895 + 3.89529i) q^{43} -10.7985i q^{47} +(-2.87076 - 4.97231i) q^{49} -4.14771 q^{53} +(-1.46410 - 2.53590i) q^{55} +(0.549538 + 0.317276i) q^{59} +(-2.35387 + 4.07702i) q^{61} +(1.81381 - 9.23490i) q^{65} +(-12.4372 + 7.18062i) q^{67} +(7.45204 + 4.30244i) q^{71} +9.94462i q^{73} +1.25847 q^{77} +13.5970 q^{79} +5.75637i q^{83} +(-12.5569 - 7.24974i) q^{85} +(-12.4235 + 7.17272i) q^{89} +(3.04743 + 2.65956i) q^{91} +(2.18825 - 3.79016i) q^{95} +(7.80107 + 4.50395i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{7} - 6 q^{11} + 6 q^{13} - 6 q^{19} - 2 q^{23} - 20 q^{25} + 8 q^{29} - 16 q^{35} - 24 q^{37} - 12 q^{41} + 6 q^{43} + 2 q^{49} - 20 q^{53} + 16 q^{55} - 18 q^{59} - 4 q^{61} - 14 q^{65} - 42 q^{67}+ \cdots + 30 q^{97}+O(q^{100}) \) 8 * q - 6 * q^7 - 6 * q^11 + 6 * q^13 - 6 * q^19 - 2 * q^23 - 20 * q^25 + 8 * q^29 - 16 * q^35 - 24 * q^37 - 12 * q^41 + 6 * q^43 + 2 * q^49 - 20 * q^53 + 16 * q^55 - 18 * q^59 - 4 * q^61 - 14 * q^65 - 42 * q^67 + 54 * q^71 + 60 * q^77 + 16 * q^79 + 6 * q^85 + 18 * q^89 - 46 * q^91 - 16 * q^95 + 30 * q^97

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